Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction

1.   Introduction

Lattice differential equations (LDEs) are systems of ordinary differential equations with a discrete spatial structure, which can naturally arise in various fields, such as image processing, neural networks, patten recognition and chemical reaction theory. These can be seen in [9,14,29,37] and the references therein. Recently, there is a particular interest on studying the species population living in a patchy environment consisting of all integer nodes, see [7,8,34,35].

Inspired by Bates [1], Chow [9], Weng et al.[34] and many other excellent survey papers, the authors in [7] considered a single-species population with two-age classes distributed over a patchy environment consisting of all integer nodes of a 2D lattice and derived the following system:

where $w_{k, j}(t)$ denote the densities of matured population in the (k, j)-th patch and time $t\geq 0$, $D_m$ and $d_m$ represent the diffusion coefficient and the death rate of the matured population, respectively, $r>0$ is the maturation time of species. $\mu = e^{-\int_0^r d(z)dz}$ and $\alpha = \int_0^r D(z)dz$ represent the impact of the death rate of immature and the effect of the dispersal rate of immature on the mature population, respectively, where $d(z)$ and $D(z)$ are the death rate and diffusion rate of the immature population at age $z\in (0, r)$, respectively. While

for any $l \in \mathbb{Z}$ and satisfy:

$\bf(i):$ $\beta _\alpha (l) = \beta _{\alpha}(\mid l\mid )$, $\gamma _\alpha (l) = \gamma _\alpha (\mid l\mid )$ for $l\in \mathbb{Z}$, that is $\beta _\alpha (l), \gamma _\alpha (l)$ is isotropic function for any $\alpha \geq 0$;

$\bf(ii):$ $\frac 1{2\pi }\sum\limits_{l = -\infty }^\infty \beta _\alpha (l) = 1$, $\frac 1{2\pi }\sum\limits_{l = -\infty }^\infty \gamma _\alpha (l) = 1$;

$\bf(iii):$ $\beta _\alpha (l)\geq 0$, $\gamma _\alpha (l)\geq 0$ if $\alpha = 0$ and $l\in \mathbb{Z}$; $\beta _\alpha (l)>0$, $\gamma _\alpha (l)>0$, if $\alpha >0$ and $l\in \mathbb{Z}$.

$\bf(iv):$ $\frac{1}{2\pi}\sum\limits_{l = -\infty}^{\infty}\beta _\alpha (l)e^{\lambda l \cos\theta} = e^{2\alpha(cosh(\lambda\cos\theta)-1)}$, $\frac{1}{2\pi}\sum\limits_{l = -\infty}^{\infty}\beta _\alpha (l)e^{\lambda l \sin\theta} = e^{2\alpha(cosh(\lambda\sin\theta)-1)}$

Assume the birth function $b$ satisfies the following assumptions:

$\bf(H1):$ $b\in C^2(\mathbb{R}^+)$ and $b(w)\leq b^{\prime }(0)w$ for $w\in \mathbb{R}^{+}$;

$\bf(H2):$ $\mu b(w) = d_m w$ has only two real roots 0 and $w^+$, and $b$ is non-decreasing on $[0, w^+]$;

$\bf(H3):$$0<b'(w^+)<\frac{d_m}{\mu}<b^{\prime }(0)$;

$\bf(H4):$ For $w\in (0, w ^+)$, $\mu b(w)>d_m w$, $b'(w)\geq 0$ and $b''(w)\leq 0$.

The authors [7] studied the existence of the asymptotic speed of propagation, the existence of monotone traveling waves and the minimal wave and its relation with the asymptotic speed of propagation. Recently, Xu [35] further showed that for any given admissible speed, all the wave profiles propagating toward a fixed direction of (1.1) have the same asymptotic behavior when they approach the limiting states, which plays a very important role in the stability of traveling waves. Meanwhile, in the past decades, there are various surveys focusing on the existence, uniqueness, asymptotic behavior and stability of traveling waves for LDEs and its continuum RDEs([1,2,3,4,6,7,8,9,11,13,16,17,19,20,21,22,37,38,39]).

In this paper, we are concerned with the stability of traveling waves of (1.1) under the assumptions $(H1)-(H4)$. In view of the symmetry, we only consider the case $\theta\in [0, \frac{\pi}{2}]$. For fixed $\theta\in [0, \frac{\pi}{2}]$ such that $\tan\theta\in \mathbb{Q}$ and the Cauchy problem (1.1)with initial data

where

we prove that the global solution $\{w_{k, j}(t)\}_{k, j\in\mathbb{Z}}$ of (1.1) and (1.2) converges exponentially to a traveling wavefront $\phi(k\cos\theta+j\sin\theta+ct)$ for $c>c_*$ (which is the minimal wave speed); while for $c = c_*$ the global solution converges to the traveling solution $\phi(k\cos\theta+j\sin\theta+c_*t)$ algebraically in time, when the initial perturbation around the wave.

As we know, many techniques are developed to investigate the stability of traveling waves such as the spectral analysis method, the weighted energy method ([5,8,12,14,15,18,23,24,25,26,27,28,30,37,38]) and the comparison principle combining the squeezing technique ([4,6,19,20,21,22,31]). Recently, Mei et al. [25] and Huang et al.[15] obtained the global stability of traveling wave fronts with noncritical speed and critical speed of nonlocal reaction-diffusion equations via the weighted energy method together with the comparison principle and Green function or Fourier transform. Zhang [38] applied this method to nonlocal LDEs in one dimension. However, there seems to be not much progress on the stability of traveling waves of system (1.1).

Particularly in [8], we only consider the case that the immature population is non-mobile, that is $D(a) = 0$ for $0<a<r$. In this case $\alpha = 0$, (1.1) reduces to

In monostable case, using weighted energy method, we derived that the Cauchy problem of (1.4) converges to a traveling wavefront when the initial perturbation around the wave is suitably small in a weighted norm. Due to the limitation of the key inequality, the result only holds for $c>\tilde{c}>c_*$.

The outline of this paper is as follows: in Section 2, we introduce some preliminaries, recall the result on the existence of traveling wave fronts of (1.1) and present our main result. Section 3 is devoted to the global stability of traveling wave fronts by using the weighted energy method combined with the semi-discrete Fourier transform.

2.   Preliminary and main result

Notations. Let $T>0$ be a number and $X$ be a Banach space. We denote by $C([0, T];X)$ the space of the $X$ valued continuous function on $[0, T]$, and by $L^1([0, T];X)$ the space of the $X$ valued $L^1$ functions on $[0, T]$. $C>0$ denotes a generic constant, while $C_k (k = 1, 2, \cdots)$ represents a specific constant. $l^{\infty}$ is the Banach space:

let $l^1$ denote the Banach space:

and denote by $l^2$ the Hilbert space

Further, $l^p_w (p\geq 1)$ denotes the weighted $l^p$-space for a weight $0<w(x)\in C(\mathbb{R})$ with the norm

For any $v = \{v_{k, j}\}_{k, j\in\mathbb{Z}}\in l^2$, its semi-discrete Fourier transform ([10,32]) is defined as

and the inverse Fourier transform is given by

where $\mathbf{i}$ is the imaginary unit.

A traveling wave of system (1.1) is $w_{k, j}(t) = \phi(x)$, $x = k\cos\theta+j\sin\theta+ct$ satisfying the following equations:

where $c>0$ is called the wave speed, $\theta$ is the direction of propagation and $\phi$ is the wave profile, subject to the boundary condition

Denoting the characteristic equation at the trivial equilibrium $w^0 = 0$ by $\Delta (\lambda , c;\theta)$, we obtain

It is easy to see that $\Delta(\lambda, c;\theta)$ is well-defined and satisfies the following properties.

Lemma 2.1. [7,Lemma 4.2] There exist a pair of $c_{\ast }$ and $\lambda _{\ast }$ such that

$\bf(i):$ $\Delta (\lambda _{*}, c_{*};\theta) = 0$, $\frac{\partial }{ \partial {\lambda }}\Delta (\lambda _{*}, c_{*};\theta) = 0$;

$\bf(ii):$ for $0<c<c_{*}$, and any $\lambda >0$, $\Delta (\lambda , c;\theta)>0$;

$\bf(iii):$ for any $c>c_{\ast }$, the equation $\Delta (\lambda _{\ast }, c_{\ast };\theta) = 0$ has two positive real solutions $0<\lambda _{1}(c)<\lambda _{2}(c)$, such that $\Delta (\cdot, c;\theta)<0$ in $(\lambda _{1}(c), \lambda _{2}(c))$, $\Delta (\cdot, c;\theta)>0$ in $\mathbb{R} \backslash \lbrack \lambda _{1}(c), \lambda _{2}(c)]$.

The existence of traveling wave fronts for (2.1) with the boundary condition (2.2) can be easily verified by using the monotone iteration technique combined with the sub-sup solutions, see [7].

Lemma 2.2. [7,Theorem 5.4] Assume $\text{(H1)-(H4)}$ hold. Then there exists $c_{*}>0$ such that for every $c\geq c_{*}$, (2.1) has a monotone traveling wave $U :\mathbb{R}\rightarrow \mathbb{R}$ satisfying the boundary condition $\lim\limits_{x\to -\infty } \phi(x) = 0, \lim\limits_{x\to +\infty }\phi(x) = w^+;$ For any $c \in(0, c_*)$, (2.1) has no nontrivial traveling wave solution satisfying $\phi(x)\in [0, w^+]$ for all $x \in \mathbb{R}$.

Recently, Xu [35] derived the asymptotic behavior of traveling waves of (1.1), which is the key premise.

Lemma 2.3. [35,Theorem 2.1] Assume $\text{(H1)-(H4)}$ hold. For any $c\geq c_*$, let $(c, \phi)$ be a solution of (2.1) with boundary conditions (2.2), then

for some $m\in\mathbb{R}$, where $l = 0$ when $c>c^*$ and $l = 1$ when $c = c^*$.

Now, define a weight function $\nu(x)$ as

where $\lambda_*$ is given in Lemma 2.1 and $x_*>0$ is chosen to be sufficiently large such that Eq.(3.19) hold.

We now state our main stability result in this paper.

Theorem 2.4. Assume that $\text{(H1)-(H4)}$ hold. For a given traveling wave front $\phi(x)$ of (1.1) with speed $c\geq c_*$, if the initial data satisfies $0\leq w^0_{k, j}(s)\leq w^+$ and condition (1.3) and the initial perturbation $w^0_{k, j}(s)-\phi(k\cos\theta+j\sin\theta+cs)\in C^0\left([-r, 0], l^1_\nu(\mathbb{Z}^2)\right)$ and

then the unique solution $w_{k, j}(t)$ of the corresponding Cauchy problem of (1.1) with the initial value $w_{k, j}(s) = w^0_{k, j}(s)$ exists globally and satisfies

and

Furthermore, for $0<\kappa\leq 2$, when $c>c_*$, the solution $w_{k, j}(t)$ converges to the traveling wave fronts $\phi(k\cos\theta+j\sin\theta+ct)$ exponentially,

for some constant $\mu_0$; when $c = c_*$, the solution $w_{k, j}(t)$ converges to the traveling wavefronts $\phi(k\cos\theta+j\sin\theta+c_*t)$ algebraically,

3.   Stability

We first consider the following initial value problem:

where $k, j\in \mathbb{Z}$, $t>0$ and $s\in [-r, 0]$. Since our analysis in this paper relies on the comparison principle, we now state the definition of super-sub solutions of (1.1) as follows:

Definition 3.1. A sequence of continuous differentiable functions $\{w_{k, j}(t)\}_{k, j\in \mathbb{Z}}$, $t\in [-r, l), l>0$ is called supersolution (subsolution) of (1.1) on $[0, l]$, if

Theorem 3.2. For any given function

and $w^0_{k, j}(s)-\phi(k\cos\theta+j\sin\theta+cs)\in C^0\left([-r, 0], l^1_\nu(\mathbb{Z}^2)\right),$ then (1.1) has a unique solution $W(t) = \{w_{k, j}(t)\}_{k, j\in Z}$ with $w_{k, j}\in C^{0}\left([-r, \infty ), [0, w^+]\right)$, and $w_{k, j}(t)-\phi(k\cos\theta+j\sin\theta+ct) \in C^0\left([0, +\infty), l^1_\nu(\mathbb{Z})\right)$. Furthermore, for any pair of sup-subsolution $w^+_{k, j}(t)$ and subsolution $w^-_{k, j}(t)$ of (3.1) on $[0, \infty)$ with $0\leq w^-_{k, j}(t), w^+_{k, j}(t) \leq w^+,$ $t\in [-r, \infty)$ and $w_{k, j}^-(s)\leq w^+_{k, j}(s)$, for $s\in [-r, 0]$, there hold $0\leq w^-_{k, j}(t)\leq w^+_{k, j}(t)\leq w^+$ for $t\in [0, \infty)$.

Note that (3.1) is equivalent to

where $\delta = 4D_m +d_m$,

and

Theorem 3.2 can be verified using an argument in [7,Theorem 3.1] and [33,Lemma 3.2] and we omit it here.

Let

for $s\in [-r, 0]$ and $k, j\in \mathbb{Z}$. It follows that

and

Denote $w^-_{k, j}(t)\left(w^+_{k, j}(t)\right)$ as the corresponding solutions of Eq.(3.1) with respect to the above mentioned initial data $w_{k, j}^-(s)\left(w_{k, j}^+(s)\right)$, i.e.

By the comparison principle, we have

Thus

where $x = k\cos\theta+j\sin\theta+ct$. Now, we can prove the stability of traveling wavefronts in three steps.

Step 1. The convergence of $w_{k, j}^+(t)$ to $\phi(k\cos\theta+j\sin\theta+ct)$.

According to (3.3) and (3.4), we have $z^0_{k, j}(s)\geq0,$ and $z_{k, j}(t)\geq 0.$ By simple calculation, $z_{k, j}(t)$ satisfies

where

with $\phi = \phi(x-l\cos\theta-q\sin\theta-cr)$. The properity $b''(\cdot)\leq 0$ on $[0, w^+]$ leads to

and

where $\widetilde{\phi}$ is some function between $\phi$ and $\phi+z.$ Then we get the following inequality,

Let $\bar{z}_{k, j}(t)$ be the solution of the following equation:

Then according to the comparison principle, we have

Let $z_{k, j}^*(t): = \nu(x)\bar{z}_{k, j}(t)$, then $z_{k, j}^*(t)$ satisfies

By Fourier transformation, one has

where $ω = (\omega_1, \omega_2)$. Taking the semi-discrete Fourier transform to (3.10), we obtain

where

In order to obtain the decay estimates of $z_{k, j}^*(t)$, we need the following lemma.

Lemma 3.3. Let $x(t)$ be the solution to the following scalar differential equations with delay

Then

where $k_3 = k_2 e^{k_1r}$ and $e_r^{k_3 t}$ is the so called delayed exponential function in the form

and $e_r^{k_3 t}$ is a solution to the following linear homogeneous equation with pure delay

Furthermore, it is pointed in [36] that when $k_1\geq k_2\geq 0$, there exists a constant $\epsilon_1 = \epsilon_1(r)$ with $0<\epsilon_1<1$ for $r>0$, and $\epsilon_1 = 1$ for $r = 0$, and $\epsilon_1 = \epsilon_1(r)\to 0^+$ as $r\to+\infty$, such that

In view of Lemma 3.3, the solution of (3.11) can be given as follows:

where $\mathfrak{B(\omega)} = \mathbf{B}e^{\mathbf{A}(\omega)r}$. Applying the inverse Fourier transform to (3.13), we obtain

Let

and

we first give an estimate of $P_{k, j}(t)$ in $l^{\infty}(\mathbb{Z}^2)$,

For the estimation of $\|e^{-\mathbf{A}(\xi)t}\|$, one has

where

and

Meanwhile, due to

we have

Then

Since when $c\geq c_*$,

(3.12) and (3.16) imply that

for a constant $0<\epsilon_1<1$.

Since $\frac{e^{x}+e^{-x}}{2}\geq 1$ for all $x\in \mathbb{R}$, we obtain

According to Taylor series expansion

It follows that

where $0<\kappa\leq 2$, $h$ is bounded and $h(\xi)\to 0$ as $\xi\to 0$. Hence, there exist $0<M_1<M$ and $a_0>0$ such that

and $0<\delta<1$ such that

Let $E_{a_0} = \left\{\xi\in[-\pi, \pi], |\xi|>a_0\right\}$. Then one has

where $m(E_{a_0})$ is the measure of $E_{a_0}$.

By changing variables $\sigma = \xi t^{\frac{1}{\kappa}}$, one has

for some constant $C>0$.

Then

Since $P_{k, j}(t)$ has no singularity for $t$ near zero, the term $t^{\frac{2}{\kappa}}$ can be replaced by $(1+t)^{\frac{2}{\kappa}}$, we get

The following inequality can be obtained similarly

Consequently, we get the following decay estimate

When $c> c_*$, one has $k_0>k_2$. It then follows that

where $\mu_1 = k_0-k_2>0$.

When $c = c_*$, we have

due to $k_0 = k_2$.

According to (3.9),

let $H = \{k, j\in\mathbb{Z}, k\cos\theta+j\sin\theta<x_*-ct\}$. If $k, j\in H$, then $e^{\lambda_*(k+ct-\xi_*)}\leq 1$. Thus, the following decay for $z_{k, j}(t)$ is clear.

Lemma 3.4. For any $r>0$, it holds that

$\bf(i):$ when $c>c_*$, then

for some $\mu_1>0$;

$\bf(ii):$when $c = c_*$, then

Next we will prove $z_{k, j}(t)$ decay exponentially for $(k, j)\in \mathbb{Z}^2\backslash{H}$.

Lemma 3.5. For $r>0$, it holds that

$\bf(i):$when $c>c_*$, then

where $\mu_2$ is a constant and satisfies

$\bf(ii):$when $c = c_*$, then

Proof. Let $z_{k, j}(t) = w^+_{k, j}(t)-\phi(k\cos\theta+j\sin\theta+ct)$, we have

Since $b''(\cdot)\leq 0$, we have

where $\widetilde{\phi}$ is a function between $\phi$ and $\phi+z$. Consequently,

for some positive constant $C_2$. Let

for $C_3\geq C_2 \geq \max\limits_{s\in[-r, 0], k, j\in\mathbb{Z}}|z_{k, j}^0(s)|$, and $\mu_2$ will be chosen later. Choose a large number $x_*$ such that for $x>x_*>>1$,

where $0<\epsilon_0<1$. We then obtain

by choosing $t\geq l_0r$ for sufficiently large integer $l_0>>1$ and

Hence, we have

Therefore, by the comparison principle, we can get

The proof is complete.

It then follows from Lemmas 3.4 and 3.5 that

Lemma 3.6. For any $r>0$, it holds that

$\bf(i):$when $c>c_*$, then

where $\mu_2$ is a constant and satisfies

$\bf(ii):$when $c = c_*$, then

Step 2. The convergence of $w^-_{k, j}(t)$ to $\phi(x)$.

By a similar argument as in Step 1, the proof can be done.

Lemma 3.7. For any $r>0$, it holds that

$\bf(i):$when $c>c_*$, then

where $\mu_2$ is a constant and satisfies

$\bf(ii):$when $c = c_*$, then

Step 3: The convergence of $w_{k, j}(t)$ to $\phi(x)$.

Lemma 3.8. For any $r>0$, it holds that

$\bf(i):$when $c>c_*$, then

where $\mu>0$ is a constant;

$\bf(ii):$when $c = c_*$, then

Acknowledgments

The authors thank the anonymous referee for their valuable comments and suggestions that help the improvement of the manuscript. And the first author was supported by the NSFs of China (No.11301121, No. 11772203).